The Compatibility of Differential Equations and Causal Models Reconsidered

Abstract

Weber argues that causal modelers face a dilemma when they attempt to model systems in which the underlying mechanism operates according to some set of differential equations. The first horn is that causal models of these systems leave out certain causal effects. The second horn is that causal models of these systems leave out time-dependent derivatives, and doing so distorts reality. Either way causal models of these systems leave something important out. I argue that Weber’s reasons for thinking causal modeling is limited in this domain are lacking.

Notes

Acknowledgements

I wish to thank James DiFrisco, Bruce Glymour, Valerie Racine, Marcel Weber, and two anonymous referees for various helpful suggestions on previous drafts of this paper.

Appendix: Classical Causal Inference

The causal Markov axiom implies a set of independence claims about a set of causally sufficient variables \(\mathbf{V}\) on a directed acyclic causal graph \(\mathcal {G}\). To state this, say that if \(X=Y\) or there is a directed path from X to Y, then X is a graphical ancestor or cause of Y and Y is a graphical descendant or effect of X. Then

Definition 1

(causal Markov) The joint probability distribution over a causally sufficient set of variables \(\mathbf{V}\) is causally Markov to directed acyclic causal graph \(\mathcal {G}\) over \(\mathbf{V}\) just in case each variable \(V \in \mathbf{V}\) is independent of its graphical non-descendants conditional on its graphical parents.

Conversely, the faithfulness axiom can be stated in the following way

Definition 2

(faithfulness) The joint probability distribution over \(\mathbf{V}\) is faithful to \(\mathcal {G}\) over \(\mathbf{V}\) just in case there are no conditional independencies in the joint probability distribution that are not entailed by the causal Markov axiom on \(\mathcal {G}\).

The causal Markov and faithfulness axioms allow us to, respectively, go from d-separation facts to claims of conditional independence and d-connection facts to claims of conditional dependence.

The graphical criteria of d-separation, in turn, allows us to go from \(\mathcal {G}\) to d-separation and d-connection claims about variables on \(\mathcal {G}\). All that is required to get us there is the graphical idea of a collider. A variable Z is a collider on a path between X and Y just in case the edges incident Z on the path are into Z. Now

Definition 3

(d-separation) Let \(\mathbf{X}\), \(\mathbf{Y}\), and \(\mathbf{Z}\) be three disjoint sets of variables of \(\mathbf{V}\) on \(\mathcal {G}\). Sets of variables \(\mathbf{X}\) and \(\mathbf{Y}\) are d-separated by \(\mathbf{Z}\) just in case, for all \(X \in \mathbf{X}\) and all \(Y \in \mathbf{Y}\), there is no path between X and Y on \(\mathcal {G}\) in which

1.

every collider has a descendant in \(\mathbf{Z}\) and

2.

no non-collider is in \(\mathbf{Z}\).

In other words two sets of variables \(\mathbf{X}\) and \(\mathbf{Y}\) are d-separated by a third set \(\mathbf{Z}\) just in case there is no d-connecting path between any \(X \in \mathbf{X}\) and any \(Y \in \mathbf{Y}\) on \(\mathcal {G}\).

Glymour, C. (2008). When is a brain like the planet? Philosophy of Science, 74, 330–347.CrossRefGoogle Scholar

Goldbeter, A. (1995). A model for circadian oscillations in the Drosophila period protein (PER). In Proceedings of the Royal Society of London. Part B: Biological Sciences (Vol. 261, pp. 319–324).Google Scholar